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  • So in my sphere flakes video, I joked about folding and cutting

  • space time, but then I thought, hey, why not?

  • So, how do you do that?

  • Well, when we wanted to fold and cut only space,

  • we chose a medium that takes place in space.

  • That is, static paper cut outs, or sphere sculpture.

  • But to fold and cut time, we need

  • a medium that happens over time.

  • I choose music.

  • Music has two easily recognizable dimensions.

  • One is time, and the other is pitch space.

  • Not quite the same as space-space,

  • but it's one dimensional, which makes things easier.

  • But let's not be confused with the notation.

  • There's a few things to notice about written music.

  • Firstly, that it's not music.

  • You can't listen to this.

  • Or, well, you can, but it'll be like--

  • [PAPER RUSTLING]

  • It's not music, it's music notation.

  • And you can only interpret it into the beautiful music

  • it represents.

  • Kind of like how a book is squiggles on a page

  • that your brain interprets into a meaningful story.

  • And maybe you don't understand it at all,

  • or understand just a literal surface meaning of the action.

  • Or maybe you can read deep and critically

  • into a story that's simple on the surface,

  • and get more from it than even the author intended.

  • Math is like this, too.

  • Secondly, written music represents

  • a two-dimensional space of pitch and time,

  • but only represents it.

  • Like, there's the suggestion that this is the time axis,

  • but it's not.

  • This is exactly the same as this.

  • Even though you're changing the values on the x-axis.

  • At least, in standard music notation.

  • Some more modern composers do make use of spatial notation,

  • just like some poets do intentionally

  • stretch out words or play with formatting.

  • But in standard notation, a stretched out word only

  • means your text editor is terrible at justified margins,

  • and has nothing to do with the word itself.

  • Pitch also doesn't entirely depend

  • on the notes placed on the y-axis.

  • So I'm going to use something a little closer to reality.

  • I've got this music box that plays a paper tape.

  • As you put the strip through the box,

  • it plays the punched holes.

  • Hold on, it's too quiet.

  • This is why music boxes usually come attached to wooden boxes.

  • I don't have a wooden box, maybe this nice wooden bowl.

  • Um.

  • Ah, there we go.

  • Music bowl.

  • Anyway, here distance along the strip

  • does translate directly to time, assuming constant crank speed.

  • [MUSIC PLAYING]

  • The box also has a set C major scale on one staff,

  • so pitch space is represented pretty directly.

  • You don't have to worry about sharps and flats, or the space

  • between staves, or the infinite possibilities

  • on a continuous logarithmic frequency scale.

  • The box magically ignores all notational elements,

  • except for where and when the holes are.

  • Each hole, each note, is a point on this strip of space time.

  • So now, let's fold and cut it.

  • It's easy to fold time so that it goes both forwards

  • and backwards simultaneously.

  • Then we can punch in some notes, and unfold it,

  • into a symmetric melody that goes first forwards, and then

  • backwards.

  • Or first backwards and then forwards.

  • Point is, it's reversible.

  • It sounds the same whether I play it like this--

  • [MUSIC PLAYING]

  • --or like this.

  • [MUSIC PLAYING]

  • But maybe you want to fold time into one finite chunk.

  • Then you could put two mirror lines,

  • which reflect themselves.

  • Which means folding time back and forth

  • infinitely, to get a little time chunk to cut up.

  • And sometimes cutting through infinitely

  • folded space time requires the use of power tools.

  • But then you can unfold a repeating,

  • back and forth musicy thing.

  • [MUSIC PLAYING]

  • Or you could leave time alone and fold space,

  • if that's what you're into.

  • And that sounds like this.

  • [MUSIC PLAYING]

  • Or you could do both.

  • [MUSIC PLAYING]

  • Maybe you could try folding at other angles,

  • but now you're mixing up the space and time axes.

  • And mixing up space and time is hazardous.

  • Though you could flip both space and time, which

  • is like rotating the music 180 degrees.

  • This one is especially fun, because you

  • can rotate any piece of sheet music,

  • and try playing it upside down.

  • And it's not just upside down, but backwards in time, too.

  • [MUSIC PLAYING]

  • Anyway, maybe you want patterns that aren't all mirror lines.

  • Like, what about a simple repeating melody?

  • Can we fold and cut that?

  • Yes.

  • Loop up the paper, cut, then unfold.

  • Or, if you're feeling snazzy, do the melody once,

  • then loop the paper through the box.

  • [MUSIC PLAYING]

  • This is probably the part where seasoned viewers will

  • be yelling something about Mobius strips.

  • In the first snowflakes video, I did fold up of a Mobius strip

  • and cut it.

  • OK, here's what happens.

  • The strip unravels to give you your shapes.

  • And then they repeat, but upside down.

  • And then they repeat, but flipped again.

  • It's a glide reflection.

  • And an example of how glide reflections really

  • are their own special sort of symmetry,

  • different from just a combination of reflections.

  • Which look like this.

  • And translation, which looks like this.

  • And which together, give you this.

  • So if you fold space time into a Mobius strip,

  • you get your melody, and then the inversion,

  • the melody played upside down.

  • And then right side up again.

  • And so on.

  • Or, rather than folding and cutting all of space time,

  • just cut and tape a little loop of space time

  • into a Mobius strip to be played over and over.

  • [MUSIC PLAYING]

So in my sphere flakes video, I joked about folding and cutting

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B1 中級

摺疊空間-時間 (Folding Space-Time)

  • 4 0
    林宜悉 發佈於 2021 年 01 月 14 日
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